A local council owns an underground railway system and the railway’s managers have to make a…

A local council owns an underground railway
system and the railway’s managers have to make a decision on whether to lower fares
in an attempt to increase passenger numbers. If they decide to reduce fares
they will then have to decide whether to launch an advertising campaign on
local television to increase awareness of the fare reduction. If fares remain
the same then it is estimated that there is a 0.7 probability that the mean
number of passengers carried per day over the next year will equal 20 000 and a
0.3 probability that it will decline to 15 000. The annual profits associated
with
»

A local council owns an underground railway
system and the railway’s managers have to make a decision on whether to lower fares
in an attempt to increase passenger numbers. If they decide to reduce fares
they will then have to decide whether to launch an advertising campaign on
local television to increase awareness of the fare reduction. If fares remain
the same then it is estimated that there is a 0.7 probability that the mean
number of passengers carried per day over the next year will equal 20 000 and a
0.3 probability that it will decline to 15 000. The annual profits associated
with these passenger numbers are estimated to be $3 million and $1 million, respectively.
If fares are reduced, but television advertising is not used, then it is
thought that there is a 0.6 probability that the mean number of passengers
carried will increase to 25 000 and a 0.4 probability that it will increase to
22 000. The resulting profits generated by these passenger numbers are
estimated to be $2 million and $1.7 million, respectively. Advertising the fare
reduction on television would increase the probability of an increase to a mean
of 25 000 passengers to 0.8 and reduce the probability that the mean will be 22
000 to 0.2. However, it would reduce the profits associated with these mean passenger
numbers by $0.6 million. The railway’s objectives are to maximize profit and to
maximize passenger numbers (since this brings environmental benefits such as
reduced traffic congestion).

(a) Utility functions for the mean numbers
of passengers carried and profit have been elicited from the railway’s chief
executive and these are given below. Plot these utility functions and interpret
them.

(b) The elicitation session revealed that,
for the chief executive, mean number of passengers and profit are mutually
utility independent. You are reminded that, in this case, a two-attribute utility
function can be obtained from:

u(x1, x2) = k1u(x1) + k2u(x2) +
k3u(x1)u(x2)

where k3 = 1 _ k1 _ k2.

The elicitation session also revealed that
k1 = 0.9 and k2 = 0.6, where attribute number 1 is the mean number of
passengers. Determine the policy that the railway should pursue in the light of
the above utilities and comment on your answer.

»

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